Lemma 10.37.6 (00GZ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.37: Normal rings
Lemma 10.37.6 (cite)

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Lemma 10.37.6. A principal ideal domain is normal.

Proof. Let $R$ be a principal ideal domain. Let $g = a/b$ be an element of the fraction field of $R$ integral over $R$ . Because $R$ is a principal ideal domain we may divide out a common factor of $a$ and $b$ and assume $(a, b) = R$ . In this case, any equation $(a/b)^ n + r_{n-1} (a/b)^{n-1} + \ldots + r_0 = 0$ with $r_ i \in R$ would imply $a^ n \in (b)$ . This contradicts $(a, b) = R$ unless $b$ is a unit in $R$ . $\square$

Comments (2)

Comment #5508 by Manuel Hoff on September 17, 2020 at 11:50

The same proof shows that in fact any factorial ring is normal. One just has to replace $(a, b) = R$ with the (weaker) condition that $a$ and $b$ are coprime in the sense that their greatest common divisor is $1$ . This is enough to make the conclusion.

Comment #5704 by Johan on November 15, 2020 at 20:55

Yes, this is stated and proved in Lemma 10.120.11.

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